In a phone call confirmed by three people, Mr. Bloomberg encouraged Mrs. Clinton to consider entering the 2013 mayor’s race, trading international diplomacy for municipal management on the grandest scale. She would, he suggested, be a perfect fit.

Much about the call, which occurred some months ago, remains shrouded in mystery. But Mr. Bloomberg’s overture to the former first lady highlights the level of his anxiety about the current crop of candidates, his eagerness to recruit a replacement who can rival his stature and his determination to become a kingmaker in the political arena he will soon exit.

In Mrs. Clinton, it seems, a mayor known for his sometimes unsparing critiques of those in public life sees a globe-trotting problem solver like himself.

During their conversation, Mrs. Clinton left little room for doubt: she was not interested in seeking the mayoralty, people briefed on the call said.

Even so, Mr. Bloomberg’s reaching out to her is rich with political intrigue. He has privately signaled support for the presumptive candidacy of Christine C. Quinn, the City Council speaker, who cleared a path for his third term by backing a change to the city’s term-limit law. The mayor’s political apparatus has begun to coalesce around Ms. Quinn, and she promotes herself as a leader who ran the city alongside Mr. Bloomberg.

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Michael R. Bloomberg is said to have viewed Secretary of State Hillary Rodham Clinton as a perfect fit for the mayor's job.Credit
Marcus Yam for The New York Times

In many corners of the city’s political world, Mr. Bloomberg’s eventual endorsement of Ms. Quinn has been considered a foregone conclusion, barring the entry of a big-name candidate into the field. But the mayor’s conversation with Mrs. Clinton, even after he had begun to telegraph his allegiance to Ms. Quinn, suggests that, to a degree previously unknown, his thinking has been unsettled.

“He is looking for somebody he can feel comfortable handing the reins over to,” said Hank Sheinkopf, a New York City political operative who worked on Mr. Bloomberg’s last campaign.

Spokesmen for both Mr. Bloomberg and Mrs. Clinton declined to comment, saying that, as a rule, they do not discuss private conservations. Told about the call on Monday, Ms. Quinn said, “Really?” as an elevator began to close, then added, “I don’t know anything about that.” The doors then snapped shut.

The people familiar with Mr. Bloomberg’s conversation with Mrs. Clinton characterized it as a casual but earnest discussion that touched on the futures of them both.

Mr. Bloomberg maintains close and warm relations with Mrs. Clinton, whose ascent within the New York political scene, in the early 2000s, closely tracked his own: Her tenure as a United States senator overlapped with Mr. Bloomberg’s mayoralty for seven years, a period when they spoke frequently about state and federal policy and finances.

Those familiar with their exchange over the mayor’s race say Mr. Bloomberg holds Mrs. Clinton in extremely high regard, admiring her willingness to devote four grinding years to working for a former rival, President Obama, and her effectiveness as a diplomat. (In 2008, he invited the Clintons to drop the crystal-studded ball in Times Square on New Year’s Eve, standing beside them on stage.)

He also relishes the chance to influence the course of future campaigns, in New York and beyond, a goal he has started to institutionalize recently with the formation of his own “super PAC.”

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Hillary Rodham Clinton, the retiring secretary of state, told Mayor Bloomberg she was not interested in running for the office.Credit
Jim Watson/Agence France-Presse — Getty Images

Despite his stated allergy to decision-making by poll, Mr. Bloomberg is well aware of Mrs. Clinton’s standing as a singularly popular Democrat who would upend the mayoral campaign.

Mrs. Clinton is not a resident of New York City, a requirement for incoming mayors. But she has overcome such hurdles in the past. To establish New York state residency for her 2000 run for Senate, she and her husband bought a home in Chappaqua, N.Y.

Democratic operatives and fund-raisers said the prospect of Mrs. Clinton’s seeking citywide office struck them as improbable, whether or not she ever mounts a second run for the White House.

As much as anything, they said, Mr. Bloomberg’s encouragement seemed to reflect his lofty view of the office — and himself.

“If in fact he did say that to Hillary Clinton, it’s only because he holds the position and therefore regards it as a step up from being president,” said Robert Zimmerman, a Democratic National Committee member from New York and a longtime fund-raiser for the party.

The city’s business community, of which Mr. Bloomberg is a leading member, has long fretted that after his 12-year mayoralty concludes, leadership of the city is likely to return to an elected official with little or no boardroom experience. Mrs. Clinton hardly fits the description of corporate chief executive — though she briefly served on the board of Wal-Mart Stores — but she has managed a large government bureaucracy as secretary of state, overseeing tens of thousands of employees in scores of countries.

In contemplating Mrs. Clinton’s future, Mr. Bloomberg was indirectly dealing with a challenge that has endlessly fascinated and bedeviled him: how to conduct a vibrant post-government life.

Mr. Bloomberg postponed his own version of that discussion three years ago by seeking a third term. Now he is becoming steeped in it, telling friends and colleagues that he is not entirely sure how he will spend his time after Dec. 31, 2013, when he must step down.

Michael M. Grynbaum contributed reporting.

A version of this article appears in print on December 4, 2012, on page A25 of the New York edition with the headline: Clinton for Mayor in 2013? Bloomberg Embraced the Idea, but She Did Not. Order Reprints|Today's Paper|Subscribe